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De Rham cohomology and conductors of curves. (English) Zbl 0632.14018

This paper shows that the conductor of an arithmetic surface (a two- dimensional scheme flat and proper over \({\mathbb{Z}})\) is the Euler characteristic of the torsion of the de Rham complex. The conductor appears in the (conjectured) functional equation for the Hasse-Weil zeta- function of the scheme and has an expression as the product of local terms which in turn can be expressed in terms of local contributions to the étale cohomology of the surface. The desired formula therefore gives a relation between local contributions to the de Rham cohomology and the étale cohomology of the scheme. It is proved by the intermediary of the Riemann-Roch formula and an expression of a local Chern class in terms of the local contribution to the Euler-Poincaré formula. In the course of the argument an expression of the conductor as the square root of the discriminant of the cup product of the de Rham complex truncated at degree 1 is obtained.
Reviewer: T.Ekedahl

MSC:

14F40 de Rham cohomology and algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14G25 Global ground fields in algebraic geometry
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[1] S. Bloch, Cycles on arithmetic schemes and Euler characteristics of curves , preprint, to appear in Bowdoin conference in Algebraic Geometry, Amer. Math. Soc. Proc. Symp. Pure Math. · Zbl 0654.14004
[2] W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0885.14002
[3] K. Kato, Vanishing cycles, ramification of valuations, and class field theory , preprint, University of Tokyo. · Zbl 0665.14005
[4] K. Kato, letter to the author. · Zbl 1038.81060
[5] J.-P. Serre, Corps locaux , Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Actualités Sci. Indust., No. 1296. Hermann, Paris, 1962. · Zbl 0137.02601
[6] J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures) , Sém. Delange-Pisot-Poitou, 1969/70, no. 19. · Zbl 0214.48403
[7] J.-P. Serre, article in Conference on Arithmetical Algebraic Geometry, Ed. by O. F. G. Schilling, N.Y., Harper and Row , 1965.
[8] P. Deligne, La conjecture de Weil. I , Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273-307. · Zbl 0287.14001
[9] P. Deligne, Intersections sur les surfaces regulieres , Springer Lecture Notes, vol. 340, Springer-Verlag, 1973, SGA 7 II. · Zbl 0272.14003
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